The Pell equation, by Edward Everett Whitford.

92

THE PELL EQUATION

of a note which I presented in June, 1862. This same
decomposition has made easily recognizable the possibility
of representing by means of singular moduli of elliptic
functions certain solutions of the Pell equation."
The importance of this equation will again be recognized
in determining the units of a real quadratic domain,
f( -Im). Finding the units, besides = 1, of such a domain,
is equivalent to solving
x2 - my2 = 1, if m 1 (mod 4),
and
(x+y) — m)2_ = - 1, if m 1 (mod 4),
at least for the + sign in the right member, for every
integral value of m.
One method of attack for the solution of
x2 - my2 = + 1
or of
(x+ Y - (Y) =M 1
is to seek out first an ambiguous principal ideal which is
different from (<em), and then from the square of this
principal ideal to obtain the fundamental unit e. The
solutions of the equations mentioned may always be
deduced from the solutions of
x2 - my2 = - a
and
where2 a
where a is a factor of 4m or m. The latter values are
always much smaller than those sought. This method
fails when the domain f( /m) contains only the ambiguous
principal ideal (]im). Even in this case shorter methods
than the standard continued fraction procedure may be
found.1
1 J. Sommer, "Vorlesungen iuber Zahlentheorie," p. 343, Leipzig, 1907.